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A331516 Expansion of 1/(1 - 10*x + 9*x^2)^(3/2).

%I A331516 #55 Aug 21 2025 00:09:44
%S A331516 1,15,174,1850,18855,187425,1832460,17705700,169569405,1612842275,
%T A331516 15256106778,143660483070,1347716324227,12603114069525,
%U A331516 117536416879320,1093553079352200,10153324144411065,94098595671581175,870667876141568070,8044341506669534850
%N A331516 Expansion of 1/(1 - 10*x + 9*x^2)^(3/2).
%H A331516 Seiichi Manyama, <a href="/A331516/b331516.txt">Table of n, a(n) for n = 0..1000</a>
%F A331516 a(n) = 1/2 * Sum_{k=1..n+1} 3^(n+1-k) * k * binomial(n+1,k) * binomial(n+1+k,k).
%F A331516 n * a(n) = 5 * (2*n+1) * a(n-1) - 9 * (n+1) * a(n-2) for n > 1.
%F A331516 a(n) = ((n+2)/2) * Sum_{k=0..n} 4^k * binomial(n+1,k) * binomial(n+1,k+1).
%F A331516 a(n) ~ sqrt(n) * 3^(2*n + 3) / (2^(7/2) * sqrt(Pi)). - _Vaclav Kotesovec_, Jan 26 2020
%F A331516 From _Seiichi Manyama_, Aug 20 2025: (Start)
%F A331516 a(n) = (1/4)^n * Sum_{k=0..n} 9^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A331516 a(n) = Sum_{k=0..n} 2^k * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A331516 a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A331516 a(n) = binomial(n+2,2) * A059231(n+1).
%F A331516 a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A331516 a(n) = Sum_{k=0..n} (5/2)^k * (-9/10)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)
%t A331516 a[n_] := 1/2 * Sum[3^( n + 1 - k) * k * Binomial[n + 1, k] * Binomial[n + 1 + k, k], {k, 1, n+1}]; Array[a, 20, 0] (* _Amiram Eldar_, Jan 20 2020 *)
%t A331516 CoefficientList[Series[1/(1-10x+9x^2)^(3/2),{x,0,20}],x] (* _Harvey P. Dale_, Nov 04 2021 *)
%o A331516 (PARI) my(N=20, x='x+O('x^N)); Vec(1/(1-10*x+9*x^2)^(3/2))
%o A331516 (PARI) a(n) = sum(k=1, n+1, 3^(n+1-k)*k*binomial(n+1, k)*binomial(n+1+k, k))/2;
%o A331516 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(1 - 10*x + 9*x^2)^(3/2))); // _Marius A. Burtea_, Jan 20 2020
%o A331516 (Magma) [1/2*&+[3^(n-k+1)*k*Binomial(n+1, k)*Binomial(n+k+1,k):k in [1..n+1]]:n in [0..20]]; // _Marius A. Burtea_, Jan 20 2020
%Y A331516 Column 5 of A331514.
%Y A331516 Cf. A059231, A084771, A383946.
%K A331516 nonn
%O A331516 0,2
%A A331516 _Seiichi Manyama_, Jan 19 2020